板波在週期性漸變結構中聚焦行為之研究

類比於光子晶體的頻溝與負折射現象，聲波在週期性排列的彈性複合材料中亦有類似的特性，此種複合材料，一般稱為聲子晶體。聲子晶體的頻溝及負折射現象，可應用於聲波濾波器或平面聲波透鏡，阻止特定頻率入射的聲子傳遞或產生小於繞射極限的聲波聚焦。本文以數值方法探討板波在週期性漸變聲子晶體平板結構中之聚焦現象。首先使用有限元素法 (finite element method, FEM)結合布拉格(Bloch)理論，建立週期性邊界條件以分析聲子晶體結構之頻散關係與特徵位移場。為瞭解不同模態在板中之波傳行為,文中分析最簡布履淵區邊界的頻散關係，並進一步取其等頻率面以檢視其非等向性。研究中採用光學漸變指數(Gradient-Index Optics)設計彈性波波傳的漸變結構。首先分析在不同頻率下填充率和折射指數之關係，並探討影響聚焦現象的各種參數，如板厚等。本文亦提出關於板波於漸變結構中聚焦行為之可能應用，即以此漸變聲子晶體平板結構當作板波的波束壓縮裝置，用來將較寬的板波導入狹窄的潑導中。模擬結果顯示，經由適當設計的週期性漸變結構可使反對稱板波波傳達到聚焦效果並可用於板波波束的壓縮。此波導壓縮裝置其使用的頻率範圍可達到設計頻率的23%。最後，本研究進行漸變型聲子晶體聚焦行為之模型實驗，以低頻壓電線波源激發板波，並以光學干涉儀接收不同位置的訊號，實驗結果成功證實漸變型聲子晶體聚焦之特性。

關鍵字

聲子晶體；板波；有限元素法；頻散；等平面

並列摘要

Similar to the photonic crystals, band gap and negative refraction also exist in the cases of acoustic waves in periodically arranged composites called phononic crystals and can potentially be applied to the filter and flat lens. In this thesis, we demonstrate the focusing behavior of a gradient-index phononic crystal numerically by finite element (FE) method and experimentally by ultrasonic experiments. In order to understand the propagation behavior of different modes in a plate, we analyzed the band structure, and further to determine the mode which is suitable for wave focusing by the equal frequency contour (EFC) analysis. Based on the study, the lowest anti-symmetric Lamb mode was employed due to its nearly isotropic EFC. Based on the concept of Gradient-Index optics, we designed the gradient index and calculated the relationship between the filling fraction and refraction index at various frequencies. We also discussed the effect of thickness on the focusing efficiency. Finally, we demonstrate the feasibility of using GRIN PC as a beam width compressor. A plate waveguide with periodic stubbed surface is adopted to achieve a band gap with frequency around the operating frequency of the GRIN PC. The simulated results show good performance on compressing plate waves into the entrance of a narrow waveguide.

並列關鍵字

Phononic crystal ； Lamb wave ； Finite element method ； Band structure ； Equal frequency surface

參考文獻

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2.E. Yablonovitch and T. J. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett., 63, 1950-1953, 1989.

3.H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B, 58, 10096, 1998.

4.M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B, 62, 10696, 2000.

5.C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104, 2002.

國際替代計量

板波在週期性漸變結構中聚焦行為之研究

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